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minimum speed required - Excellent question

I have a question which I have been unable to solve for quiet a time now.. I would be really grateful if someone could tell me the proper solution to this problem.. The problem goes like this -- What is the minimum speed with which a person can throw a ball such that it crosses a long cylindrical object of radius R (placed with its axis perpendicular to plane of motion of the ball)?? Ignore the height of the person throwing the ball

Note by Yash Dalmia
4 years ago

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Given the radius of the cylinder to be R. A Ball thrown can travel the maximum distance before touching the lower half of the cylinder if thrown at an angle of $$\frac{pi}{4}$$ or 45 degree. Equating energy equation

$$m.g.h = \frac{1}{2}.m,v^{2}$$

$$h = 2.R$$ as the ball first travels up until it reach the top most periphery and then travels down until it reach the lowest periphery

The component of the velocity which causes the ball to rise if $$v.\sin \theta$$

so $$m.g.h = \frac{1}{2}.m,v^{2}$$

or $$g.h = \frac{1}{2}.{v.\sin \theta}^{2}$$

or $$v = \frac {\sqrt {2.g.h} }{\sin \theta}$$

or $$v = \frac {\sqrt {2.g.h} }{\frac{1}{2}}$$

or $$v = 2.\sqrt {g.h}$$

as $$h = 2.R$$

so $$v = 2.\sqrt {g.2.R}$$

or $$v = 2.\sqrt {2.g.R}$$ · 4 years ago